Alexander Gnedin
The boundary problem is considered for inhomogeneous increasing random walks on the square lattice ${\mathbb Z}_+^2$ with weighted edges. Explicit solutions are given for some instances related to the classical and generalized number triangles.
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We analyse the asymptotic behaviour of the probability of observing the expected number of successes at each stage of a sequence of nested Bernoulli trials. Our motivation is the attempt to give a genuinely frequentist interpretation to the notion of probability based on finite sample sizes. The main result is that the probabilities under consideration decay asymptotically as $n^{-1/3}$, where $n$ is the common length of the Bernoulli trials. The main ingredient in the proof is a new fixed-point theorem for non-contractive symmetric functions of the unit interval.
We consider inhomogeneous Bernoulli measures of the form $\prod_{x\inΛ} p_x$ where $p_x$ are prescribed and uniformly bounded above and below away from 0 and 1. A comparison inequality is proved between the Kawasaki and Bernoulli-Laplace Dirichlet forms. Together with a recent result of Caputo on the gap of the Bernoulli-Laplace model, this proves a spectral gap of the correct order L^{-2} on cubes of side length L for the Kawasaki dynamics. The two block estimate of hydrodynamic limits is also obtained.
On Bernoulli trials with unequal harmonic success probabilities
本文研究成功概率为不等调和数的伯努利方案，涵盖成功次数、等待时间及渐近性质，并探讨其与Sibuya和Yule-Simon分布的关系，应用于可靠性、物种抽样等领域。
The longest stretch $L(n)$ of consecutive heads in $n$ i.i.d. coin tosses is seen from the prism of large deviations. We first establish precise asymptotics for the moment generating function of $L(n)$ and then show that there are precisely two large deviation principles, one concerning the behavior of the distribution of $L(n)$ near its nominal value $\log_{1/p} n$ and one away from it. We discuss applications to inference and to logarithmic asymptotics of functionals of $L(n)$.
Suppose you can color $n$ \emph{biased} coins with $n$ colors, all coins having the same bias. It is forbidden to color both sides of a coin with the same color, but all other colors are allowed. Let $X$ be the number of different colors after a toss of the coins. We present a method to obtain an upper bound on a median of $X$. Our method is based on the analysis of the probability distribution of the number of vertices with even in-degree in graphs whose edges are given random orientations. Our analysis applies to the distribution of the number of vertices with odd degree in random sub-graphs…
Generalization of Arithmetico-Geometric Series and the Expectation Value of a $k$-Run of a Bernoulli Trial
本文通过算术几何斐波那契级数与递推关系，对伯努利试验中首次出现k个连续成功所需试验次数的期望值进行了新颖推导，虽结果非新但推导方式有创新性。
The standard textbook method for estimating the probability of a biased coin from finite tosses implicitly assumes the sample sizes are large and gives incorrect results for small samples. We describe the exact solution, which is correct for any sample size.
Feller (1945) provided a coupling between the counts of cycles of various sizes in a uniform random permutation of $[n]$ and the spacings between successes in a sequence of $n$ independent Bernoulli trials with success probability $1/n$ at the $n$th trial. Arratia, Barbour and Tavaré (1992) extended Feller's coupling, to associate cycles of random permutations governed by the Ewens $(θ)$ distribution with spacings derived from independent Bernoulli trials with success probability $θ/(n-1+θ)$ at the $n$th trial, and to conclude that in an infinite sequence of such trials, the numbers of spacing…
In this note we establish a uniform bound for the distribution of a sum $S_n=X_1+\cdots+X_n$ of independent non-homogeneous Bernoulli trials. Specifically, we prove that $σ_n \mathbb{P}(S_n\!=\!j)\leqη$ where $σ_n$ denotes the standard deviation of $S_n$ and $η$ is a universal constant. We compute the best possible constant $η\sim 0.4688$ and we show that the bound also holds for limits of sums and differences of Bernoullis, including the Poisson laws which constitute the worst case and attain the bound. We also investigate the optimal bounds for $n$ and $j$ fixed. An application to estimate t…
We consider a random trial-based telegraph process, which describes a motion on the real line with two constant velocities along opposite directions. At each epoch of the underlying counting process the new velocity is determined by the outcome of a random trial. Two schemes are taken into account: Bernoulli trials and classical Pólya urn trials. We investigate the probability law of the process and the mean of the velocity of the moving particle. We finally discuss two cases of interest: (i) the case of Bernoulli trials and intertimes having exponential distributions with linear rates (in whi…
We prove the equivalence of ensembles for Bernoulli measures on $\mathbb{Z}$ conditioned on two conserved quantities under the situation that one of them is spatially inhomogeneous. For the proof, we extend the classical local limit theorem for a sum of Bernoulli independent sequences to those multiplied by linearly growing weights. The motivation comes from the study of random Young diagrams. We discuss the relation between our result and the so-called Vershik curve which appears in a scaling limit for height functions of two-dimensional Young diagrams.
Type II success runs of Bernoulli trials separated by a gap
本文研究了带有间隔的成功连续运行的二型伯努利试验，推导了概率质量函数、最长连续运行分布及因子矩的表达式。
On a connection between total positivity and Bernoulli stopping problems
本文研究了总正性与伯努利停止问题之间的联系，证明在特定条件下我的策略是最优的，并通过示例展示了其应用。
We study an optimal investment problem with multiple entries and forced exits. A closed form solution of the optimisation problem is presented for general underlying diffusion dynamics and a general running payoff function in the case when forced exits occur on the jump times of a Poisson process. Furthermore, we allow the investment opportunity to be subject to the risk of a catastrophe that can occur at the jumps of the Poisson process. More precisely, we attach IID Bernoulli trials to the jump times and if the trial fails, no further re-entries are allowed. We show in the general case that …
This article employs the relation between probabilities of two consecutive values of a Poisson random variable to derive conditions for the weak convergence of point processes to a Poisson process. As applications, we consider the starting points of k-runs in a sequence of Bernoulli random variables and point processes constructed using inradii and circumscribed radii of inhomogeneous Poisson-Voronoi tessellations.
We propose an approach to construct Bernoulli trials $\{X_i, i\ge 1\}$ combining dependence and independence periods, and call it Bernoulli sequence with random dependence (BSRD). The structure of dependence, on the past $S_i = X_1 + \ldots + X_i$, {defines} a class of non-Markovian random walks of recent interest in the literature. In this paper, the dependence is activated by an auxiliary collection of Bernoulli trials $\{Y_i, i\ge 1\}$, called {\it memory switch sequence}. We introduce the concept of {\it memory lapses property}, which {is} characterized by intervals of consecutive independ…
Deterministic algorithms for inhomogeneous Bernoulli trials: Shapley value of network devices
针对非齐次伯努利试验下的网络设备博弈，提出线性或二次时间的确定性Shapley值近似算法，并给出严格误差分析，同时用定积分与组合方法补充了Shapley原始证明。
Gibbs Properties of the Bernoulli field on inhomogeneous trees under the removal of isolated sites
研究度有界的非正则可数无限树上 i.i.d. Bernoulli 场，证明移除孤立 1 的确定性变换在大 p 时破坏拟局部 Gibbs 性，而小 p 时仍存在拟局部 specification。
Using the Blackwell-Cover Guessing Strategy to Separate Unobserved Bernoulli Trials
本文研究Blackwell-Cover猜测策略的变体，用于分离来自两个均匀分布的未观测伯努利试验，证明在某些情况下可将未观测硬币按正面概率均值分离，且递归应用可几乎完全分离两类硬币。
On a conjecture of Lamkin and Tkocz: Log-convexity of moments of Bernoulli sample means
研究伯努利样本均值矩的对数凸性，通过基于\(2n\)次试验中成功总数的证明方法，将不等式简化为超几何随机变量相关的凸序比较，解决了拉姆金和托茨猜想的伯努利情形。
Given a $p$-coin that lands heads with unknown probability $p$, we wish to produce an $f(p)$-coin for a given function $f: (0,1) \rightarrow (0,1)$. This problem is commonly known as the Bernoulli Factory and results on its solvability and complexity have already been obtained. Nevertheless, generic ways to design a practical Bernoulli Factory for a given function $f$ exist only in a few special cases. We present a constructive way to build an efficient Bernoulli Factory when $f(p)$ is a rational function with coefficients in $\mathbb{R}$. Moreover, we extend the Bernoulli Factory problem to a…
The Last-Success Stopping Problem with Random Observation Times
本文研究随机观测时间下最大化在最后一次成功时停止的概率问题，利用高斯超几何函数证明短视策略在 ν≥θ 时最优，并给出获胜概率公式及大 N 极限形式。
Non-intersecting path constructions for TASEP with inhomogeneous rates and the KPZ fixed point
本文研究了具有不均匀速率的TASEP模型，利用非相交路径和边值问题，推导出相关核的表达式，推广了KPZ固定点理论。
Mixing cutoff for simple random walks on the Chung-Lu digraph
本文研究Chung-Lu有向图上简单随机游走的混合行为，证明在平均度随图大小对数增长的弱稠密状态下，总变差距离在熵时间处呈现截断行为且收敛于高斯尾函数，将已有硬约束模型的结果扩展至软约束模型。
Variational principles for Hausdorff and packing dimensions of fractal percolation on self-affine sponges
本文建立了一类统计自仿海绵（包括Barański和Gatzouras-Lalley地毯与海绵的分形渗透集）的Hausdorff维数与packing维数的变分原理，通过计算非齐次Mandelbrot测度的维数并利用局部化数字频率构建有效覆盖，获得了维数的精确上界。
Sharp spectral norm concentration and regularization of sparse random tensors
本文证明稀疏随机张量谱范数的尖锐集中不等式，去除对数因子，并推广至非齐次采样及随机超图模型。
We consider the $N$-particle noncolliding Bernoulli random walk --- a discrete time Markov process in $\mathbb{Z}^{N}$ obtained from a collection of $N$ independent simple random walks with steps $\in\{0,1\}$ by conditioning that they never collide. We study the asymptotic behavior of local statistics of this process started from an arbitrary initial configuration on short times $T\ll N$ as $N\to+\infty$. We show that if the particle density of the initial configuration is bounded away from $0$ and $1$ down to scales $\mathsf{D}\ll T$ in a neighborhood of size $\mathsf{Q}\gg T$ of some locatio…
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